Tuesday, June 16, 2009

What should philosophers of science know about philosophy of mathematics?

I am working on a survey article on "Philosophy of Mathematics" for the forthcoming Continuum Companion to Philosophy of Science, edited by French and Saatsi. In the article I aim to give philosophers of science some orientation on the debates within philosophy of math that can be intimidating to the uninitiated. But I would also like to highlight some topics that clearly fall into a common area between philosophy of science and philosophy of mathematics, but which are typically pursued in isolation by one specialist or another. Here I am thinking of debates about the role of mathematics within science and what this tells us about mathematics and/or science. A central instance of this is the indispensability argument for platonism about mathematics. While this has been an ongoing debate in philosophy of mathematics for 30 years, the debate is not informed by contemporary philosophy of science. See, for example, the ongoing debate about mathematical explanation of physical phenonomena. At the same time, philosophers of science seem to not have much to say about (i) whether mathematics is indispensable to our best science in some interesting sense or (ii) what this might show about science if it is true.

Any suggestions for other debates in this common area of interest between philosophy of science and philosophy of mathematics? Any suggestions for how to close this gap between two areas of contemporary philosophy?

10 comments:

This is tangential to your question, but it's always struck me as odd that there has not been more interaction between philosophers of science and philosophers of mathematics on the foundations of probability. Probability seems like an interesting and useful case study for different views in the philosophy of mathematics: for instance, various 'objective' interpretations of probability seem to fit the mold of a Maddy-style naturalistic platonism; De Finetti style reduction of all probability to subjective probability smells of intuitionism or constructivism). At the same time, the philosophy of math would seem to suggest some options that have not been explored in discussions of the foundation of probability (e.g. formalism or fictionalism).

for a while now, maybe with Lakatos' "Proofs and Refutations" as the starting point, there are two streams in philosophy of mathematics (see, e.g., the upcoming conference Two Streams in Philosophy of Mathematics). In a nutshell: The traditional one focuses on the notion of formal proof as being essential for mathematics, and thus emphasises one aspect in which mathematics seems to be fundamentally different from science. The other one, dubbed `marverick', focusses on mathematical practice. The latter, I think, points at similarities between scientific and mathematical reasoning. Once one gives up the idea that mathematicians mainly develop formal proofs from some given set of axioms, many differences between mathematical and scientific activities disappear. Of course, scientists are interested in phenomena occurring in the real world, but once a scientist has gathered her data and gets back in the office to develop her theory, her work isn't that different of a mathematician who is trying to develop a theory about some new conjecture or mathematical observation. It is no surprise then, that people like Hilbert would lecture both in mathematics and physics. Model-based reasoning and analogical reasoning have become more and more important in accounts of scientific reasoning (e.g., Nersessian's "Creating scientific concepts" (2008)), and there is no reason why these shouldn't play a role also in the activities of mathematicans. For instance, Grosholz explictly discusses mathematics and science together in her "Representation and productive ambiguity in mathematics and science" (2006), in which she brings examples from chemistry, biology, geometry, and logic.

Thus, I think the historical turn that philosophy of science has taken with Kuhn and the developments in philosophy of mathematics since Lakatos, both of which put the actual practice at the center of attention, have indeed much common ground; and there is still much left to be explored here.

Thanks for the suggestions. On probability, I guess the idea is that we can consider views developed for mathematics and see if they can be used to handle problems interpreting probabilities? There is a further possibility of using probability theory within mathematics, especially for the field of "experimental mathematics" (Baker has an article on this.)

I am unsurprisingly most focused on issues of the applicability of mathematics in science. Still, this issue is very vague and depending on how it is clarified I think it could be of more or less interest to philosophers of science. For example, I think philosophers of science are not that interested in indispensability questions because the ultimate issue is platonism. But an attempt to say what the positive contribution that mathematics makes to science should be of wider interest (I hope!).

Dirk, I agree, that we should look for similarities between mathematics and science, and these sorts of conceptual issues are clearly a good place to start. Of course, there are also differences between mathematics and science, but I expect even these differences can be useful for philosophers of science to think about.

Seamus, thanks for the suggestion. Someone else has already pointed out that there is no reference to Wigner, and I will certainly add in a footnote or new paragraph in the final draft. Still, I think that Wigner's argument has a bad reputation in philosophy of mathematics. The best attempt to make good sense of it is Steiner's 1998 book, and while I have spent a lot of time struggling with this book, I think it had a small impact, even in debates about applicability.

Chris, not surprisingly, I have a lot to say about mathematical structuralism, both as it relates to philosophy of mathematics and as it relates to philosophy of science, the latter, in particular, to recent structural realism debates. As regards philosophy of mathematics, I think Dirk’s distinction between the two streams of “proof” and “practice” is both important and significant. Though, I think it might be helpful to use a more general term like ‘justification’ instead of ‘proof’. I think too it’s a distinction that is not often appreciated in the philosophy of science, especially in its borrowing of the various “isms” that one finds in the philosophy of mathematics. Minimally one requires an argument for using an “ism”, that was intended in the stream of justification in the philosophy of mathematics, now used, by the philosopher of science, in the stream of practice. Think, for example, of the logical positivists’ problematic attempts to use logicism/formalism to frame “the received view" of scientific theories. Consider too that if one sees the stream of “justification” as more in accordance with foundationalist positions in the philosophy of mathematics, then structuralism (framed either with set theory or with category theory) appears as a strong position both historically and philosophically. And when one considers the practice stream it seems here (and again, perhaps not surprisingly that I would argue this) that category theory is a better framework, since it can be used to frame what we say about various mathematical structures that are used both by the mathematician and by the scientist. For example, Resnik restricts himself to categorical (in the logic sense of all models are isomorphic) structures and Shapiro to what he calls non-algebraic structures. Likewise, Maddy’s naturalism restricts itself to classical (bivalent) structures. In setting of category theory, no such restrictions are made. Moreover, the very use of the term ‘foundation’ (as a tool of organization) by category theorists is as much in line with the practice stream as it is with the justification stream. Getting back, then, to my point of the philosopher of science needing an argument for moving an “ism” from one stream to another. Besides the point of whether category or set theory is a better frame for either stream, there also remains the issue of philosophers of science (following in the footsteps of Suppes and/or Bourbaki) presuming that the semantic view of scientific theories, in so far as it talks of models as structures, needs itself be framed by set theory. I have argued that it need not and, moreover, that the notion of structure (or shared structure) ought not be so framed when one turns to run structural realist arguments. One last point, in the structural realism literature, the structure/nature debate is sometimes problematically run together with the form/content debate in philosophy of mathematics and this with a quite confused resulting view of mathematical structuralism; as we ought have learned from Benacerraf, for the matheamtical structuralist, talk of the nature of objects is abandoned. So if the philosopher of science wants to talk about the nature of objects, then, if he uses mathematical structuralism, he has simply used the wrong “ism”. But, even if one sticks to structuralism, this does not mean, contra the ontic structural realist, that objects do not have a nature (that all there is is structure), it means that, IF mathematics is the language of nature (and here’s where Wigner comes to play a crucial role), then objects have no nature. But that’s a big IF, and one that cries out (if not screams out with lots of foot stamping) for an argument.

Elaine, thanks for your comment -- this could be a whole post in itself, and might be useful to get more dialogue between structuralists in different areas.

My rough sense of it is that the most interesting advocates of category theory, such as Awodey and yourself, are trying to get past traditional foundational debates about mathematics, rather than simply replace set theory by category theory as a traditional foundation. I need to think more about how to note this, or at least direct interested readers of my survey to some relevant papers!